(Another) Mathematician’s Lament – How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form

“A brilliant research mathematician who has devoted his career to teaching kids reveals math to be creative and beautiful and rejects standard anxiety-producing teaching methods. Witty and accessible, Paul Lockhart’s controversial approach will provoke spirited debate among educators and parents alike and it will alter the way we think about math forever.” Source: Bellevue Literary Press

As a mathematics teacher and long-time student of mathematics, I was overjoyed when I came across this book. Finally, I thought, an ode to the profound beauty and elegance of this most precise and direct human languages. And, hopefully, an expose on the state of mathematics education, and a plea to change course, maybe even some practical suggestions on how we may begin to do this.

Lockhart and I started in lock-step. YES. The current state of mathematics education is a TRAVESTY – we are most emphatically not teaching students to fully appreciate its abstractive powers, philosophical implications, and inherent structure. Instead, we are taking the artifacts of doing mathematics and positing that these are what math is about.

An example: numbers.
My absolute favorite. As a math teacher, I get this a lot on my first day of a new school year, “Ms. S., what 36*129?” I am, of course, expected to do this quickly without a calculator. Since, unfortunately, most students have been taught to think that math is about numbers, or shapes, or equations, or graphs, or in general, mathematical objects.

Mathematician Paul Halmos (1916-2006) wrote an excellent essay on this misconception. An excerpt:

“to begin with, mathematicians have very little to do with numbers. You can no more expect a mathematician to be able to add a column of figures rapidly and correctly than you can expect a painter to draw a straight line or a surgeon to carve a turkey-popular legend attributes such skills to these professions, but popular legend is wrong. There is, to be sure, a part of mathematics called number theory, but even that doesn’t deal with numbers…” PDF File here (YES! What number theory is about, by the way, is the concept of counting; numbers are but an artifact…)

I’ve never been a human-computer type of whiz-kid, and I can’t compute ‘in my head’, without the aid of pen and paper. I don’t, in fact, even like numbers particularly, and thankfully never had to deal with them much in school (as a math major). I’m not even a quantitative type of person. And I certainly do not believe in the applicability of mathematics to all human endeavors (well, I believe it can be applied in every circumstance, but not with positive effects; take, for example, the current testing regime in education as a prime example of the devastation that can be wreaked by our belief in numbers to solve all problems).

But, I still love math. And I choose to prove my teacher worthiness by beginning the year-long conversation with my students about how math is not about its objects, but is rather a language. A language unlike any other we speak, one predicated on conciseness, precision, and directness. A language that may also be applied widely, but with caution and attention to ethical and social implications.

But let me get back on task. So I absolutely agree with Lockhart: mathematics, as language, is an art. It is indeed a tragedy that in too many schools today, math education consists of futile exercises in computation, memorization of formulas, of solving contrived word problems, and, more recently, manifests as an endless quest to eliminate wrong answer choices on standardized tests. YES of course I agree: math should not be taught in procedural fashion, formulas should not be blindly memorized, problems should not be contrived to be about “real life” situations.

I’m also in agreement that we need to invest in programs that will train all of our math teachers in formal mathematics. At the moment, most math teachers in the US have transitioned from the workforce (engineering mostly, some physics and other sciences) or have degrees in math education. It is my and Lockhart’s contention that a deep understanding of the subject is required in order to be able to relate the essence of math.

[In an ironic twist, badly applied quantitative measures of unquantifiable phenomena (such as the experience of student learning) suggests that math degrees don’t make a difference in terms of student “success” (See this Edweek Article)].

While I agree with Lockhart’s assessment of the inadequacy of the current state of math education, I strongly dissent to his suggestions for how we should move towards reform. A Mathematician’s Lament lacks any kind of historical understanding, and does not at all consult pedagogical and curriculum literature.

For example, Lockhart writes that “word problems” should not be contrived to be about real life (I agree with this point), but then he continues that mathematics is beautiful precisely because it is irrelevant to ‘real life’..

I cannot comprehend how another mathematician could possibly believe the beauty of mathematics comes from the “irrelevance” of its abstractions: in fact, the reason math is SO powerful is that these abstract representations have all been historically “discovered” or “invented” (depending on what you believe math is: inherent in the world, or a human game of abstraction)–particularly in order to try to model and explain phenomena observed in “the real world.”

Lockhart says math was created by humans “for their own amusement” (p. 31), but ignores that in fact all branches of mathematics in the past were created in response to actual world problems, and not only that, but now, some of the most fascinating mathematics is being created again in response to solving some of the most complex problems we have imagined, such as the mathematics behind string theory. I don’t know how Lockhart could possibly consider that humans invented counting, ways to measure their plots of land and keep track of money, or ways to measure the orbits of planets (thus leading us to the current “space age”) as “purely amusement”–perhaps, if life is amusement in general, but really, all of these inventions had a very real, concrete, specific historical cultural purpose and are not “just made up” for fun!!

I teach functions (precalculus, AP calculus) and the main theme is how basically, in life, we track patterns of change in anything and everything–public health data, unemployment, polling, the stock market, baseball stats, etc. Functions are just the most abstract way to represent these changing patterns over time (or some other variable) and thus give us the powerful tool of projecting into the future/past and otherwise analyzing trends. Yes, functions are abstract, but they are not “just fantasy play,” irrelevant to the real world, or made up simply for the fun of it, in fact, quite the opposite of all of these.

My (and I believe, many) students would be aghast to learn that someone is suggesting an overhaul of math education based on the idea that “kids don’t really want something that is relevant to their daily lives.” This is the most absurd statement I have ever read so I am guessing Lockhart knows nothing about adolescent/child development, interest, and pedagogical literature. Learning in general is based on making connections to prior knowledge, and I have never heard any question asked more often in math class when I didn’t explain the relevance in advance than “Why do I need to know this? How is this relevant to my life?” This is probably the MOST pressing question for adolescents in general…* (See very long note on Dualism in Lockhart & Real Life Applications)

Other examples of pedagogical tragedies in this book include Lockhart’s admonitions that “you can’t teach teaching,” that “schools of education are a complete crock” and that teachers shouldn’t lesson plan because this is somehow “not real” or authentic (p. 46-47). While I agree schools of education are not preparing our teachers well and what we need is much more systemic training in content knowledge, it is absolutely not supported by any peer-reviewed research that teaching is something you “have” that you don’t need to “learn” and, further, that you shouldn’t plan because this is inauthentic.

A plan should of course never prevent a teacher from moving in new directions as suggested by the course of the class, but coming in without a plan is certainly not considered sound practice in any theory of learning and from any angle, and in general is not a sound principle of life (i.e., just doing everything by the seat of your pants and counting on your “genius” to lead you through whatever you should have planned usually doesn’t work, unless you are in a feel-good movie). Only in Lockhart’s fantasy “lala land” of irrelevancy is planning a vice and not a virtue. Plus, there’s so much more to “planning” than thinking about the flow of the lesson, how you will help students make connections, etc. I assess and plan hand in hand for example, and I tailor my classes for my particular students that year.

*Very Long Note on Dualism in Lockhart & Real World Applications
Math is a language, and as such, art, that captures the most abstract essence of our world. Some even go as far to say that math is a structure of our very universe (see Penrose and Tegmark); I’m trudging through their work at the moment and am not convinced, but this may change.

Rather, what I’ve always believed is that math is embodied in our cognitive schemas and perception, and that this is precisely what makes it so wonderful: humanity’s inherent capacity for thinking about the real world in this abstract way (see Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being by George Lakoff & Rafael Núñez). The point is, these ideas are born in experience: and, in turn, our experiential perceptions are shaped by these ideas, creating the cyclical process of learning and expanding our horizons.

From this perspective, in which experience/perception are perpetually interconnected to our cognitive schemas in a cycle of expansion, to say as Lockheart does, that math, or that anything, for that matter, is purely “of the mind” is basically Descartes all over again, “only the thought exists”. And we all know how well that turned out.

Now, I am not proposing the other side of the dualistic coin: teaching “math for engineering” type courses in which the emphasis is on the application. What I think is essential is to teach math in the context of its history, its applicability, its ethics, its abstractive prowess, its meaning.

Nothing is “born in our mind” alone; nothing exists in our “mind” alone; and for anything to make sense, the very idea of something having a sense, comes from our experiential perception.

Example #1: take the case of Zero (see Charles Seife’s Zero: The Biography of a Dangerous Idea). We tend to think of “0” as a number, perhaps like any other, but this is far from accurate. Zero has a complex history within that of counting, and it was not at first even considered a number, but a place holder for “nothing.” While anthropologists have discovered potential counting artifacts as old as 30,000 years, zero is only a few thousand of years old, if that. It took many tens of thousands of years after the adoption of numbers to “invent” the concept of “zero”-most likely because zero/the idea of cataloging “nothing” was not part of the daily experience of tracking items, livestock, or people. In fact, mathematicians to this day continue to refer to the set of integers “1” and above as “natural numbers”, and do not include 0 in this set.

Example #2: the concepts of positive and negative, the number line as a construct. The number line parallels our perceptive ability to set dualistic reference points in/with our bodies, such as east-west, up-down, right-left, and so on; this reference-setting tendency is further related to our bipedal structure. Of course, we also think in terms of “continuums”, mostly one-dimensional (linear). I often wonder what our mathematics would be like if humans had the anatomy of octopuses!

I read this book some time ago (2008?). Posted the first review in 2010, which has been significantly edited from its original version in this April 30, 2016 update.

8 comments

To say that mathematics are “irrelevant” is one of the stupidest thing ever. They were not created for pleasure, they were created because we needed them.
I mean, does this guy know physics, biology, architecture or economy (to quote just a few)? Where would they be without maths?
Sometimes I wonder if authors actually read what they write…

I know, right?! And this is coming from a mathematician. UGH. His perspective reflects a kind of academic snobbery I’ve run into at times in math departments, for which “application” is a bad word. But you are so right – where would any of these fields be without the underlying math?
Thank you for stopping by and reading 🙂

I can tell you are very passionate about teaching. We need more teachers like you! I’m not going to lie, this topic is a little over my head, so I’m just going to trust you and agree with everything you said 😅 If I had to guess, I would have pegged you for an English or History teacher… I find you fascinating indeed!

Aw, thank you. So the story is I spent my entire life up through college and my math degree being ‘tracked’ into “STEM” type programs (STEM MS, HS, college degree)… but all the while I was dreaming of a liberal arts degree. So now that my time is my own, I’m making up for 17-18 years of formal education. While I do love reading both literature and history, I am definitely not qualified to teach these subjects. I admire English & History teachers tremendously, I don’t know how they do it – in a math class at least one doesn’t have to worry about diverging opinions and leading debates and overcoming prejudices and all the rest.

Ah! The moment I saw the book title, I wanted to buy it and read it, but your review convinced me otherwise. As a mathematician myself, I hate it when people tell me they wouldn’t study math because “it’s too dry.” Anyone who says this has no understanding what math is all about. I couldn’t agree with you more — math is, above all else, art.

That’s why I had high hopes when I saw the book, but to say that math is irrelevant to the real world is laughable. Math is everywhere around us! During my undergrad studies I took a course in Abstract Algebra (rings, fields, etc), which is as abstract as abstract gets, and even that has numerous applications — coding, cryptography, physics, art, and can often be observed in nature. Later on, when I studied stochastic calculus, many of my fellow students wondered why we would possibly need that, but once I graduated and started work in the “real world,” I get to use it all the time!

Interestingly enough — most of the mathematicians I know either write fiction, or are involved with some other art. Another connection to creativity!

Hi Elena, thank you so much for reading and your very thoughtful response! So agree with everything you are writing – math is absolutely art – definitely not a coincidence that most mathematicians are also musicians or writers or (in my case) dancers 🙂

I am more of an analysis girl myself but I did find abstract algebra fascinating – it’s definitely the foundation of pretty much everything else (well, the second layer of the foundation, first being logic). So definitely, the applications run the gamut. Whenever people claim math isn’t useful in real life, I just want to pull out my hair because we wouldn’t have ANY science and tech we have today if it wasn’t for the math underpinning it. I’m so sick of hearing that because most people may not work with math directly, math is not useful. IT’S THE FOUNDATION OF EVERYTHING 🙂 (ok sorry. I get a little emotional ;)) Also for a mathematician to say its beauty lies in its irrelevance is simply dumbfounding to me.